Applied Ontology 0 (2007) 1 1IOS Press

Representing and Reasoning over aTaxonomy of Part-Whole Relations

C. Maria Keet ∗ and Alessandro ArtaleKRDB Research Centre, Faculty of Computer Science, Free University of Bozen-Bolzano, ItalyE-mail: {keet, artale}@inf.unibz.it

Abstract. Many types of part-whole relations have been proposed in the literature to aid the conceptual modeller to choose themost appropriate type, but many of those relations lack a formal specification to give clear and unambiguous semantics to them.To remedy this, a formal taxonomy of types of mereological and meronymic part-whole relations is presented that distinguishesbetween transitive and intransitive relations and the kind of entity types that are related. The demand to use it effectively bringsafore new requirements for automated reasoning over a hierarchy of relations. To ensure logically and ontologically correctinferencing over both the class and role hierarchy, the new reasoning service RBox compatibility for Description Logics reasonersis introduced. The proposed combination of formal semantics and the new reasoning service will improve the representation ofthe application domain when using part-whole relations in conceptual models and ontologies.

Keywords: part-whole relation, mereology, conceptual modelling, Description Logics reasoning

1. Introduction

Many ontological and cognitive aspects of the part-whole relation have been discussed (e.g. Artale etal. (1996); Bittner and Donnelly (2005); Gerstl and Pribbenow (1995); Odell (1998); Shanks et al. (2004);Varzi (2004, 2006a); Vieu and Aurnague (2005); Winston et al. (1987)) and proposals to include thisrelation to conceptual modelling and knowledge representation languages have been suggested, such as(Barbier et al. (2003); Guizzardi (2005); Motschnig-Pitrik and Kaasbøll (1999)) for UML, (Bittner andDonnelly (2005); Lambrix and Padgham (2000); Sattler (1995); Schulz and Hahn (2000)) for DescriptionLogics, Shanks et al. (2004) for ER, and Keet (2006a) for ORM. These communities introduced differenttypes of part-whole relations, which sometimes are different types of mereological parthood relations,whereas others appear to be motivated by cognitive and linguistic use of ‘part’ (meronymy). To representthe Universe of Discourse as accurately as possible, it is an imperative to be able to identify and use theappropriate part-whole relation that is closest to the real world it is supposed to represent. Being moreprecise during the conceptual analysis stage by taking into account ontological notions of part-whole re-lations will improve the quality of the conceptual model, thereby reducing errors in software develop-ment and saving resources during the testing phase, which also result in better application software. Otheradvantages obtained by representing part-whole relations more precisely range from the implementationstage by supporting product quality assurance and fault identification of e.g. a specific part-componentof a device – as one can represent the parts properly – to reasoning over conceptual models (and domainontologies) by having distinguished transitive and intransitive part-whole relations, such as classifyinghierarchies, checking concept satisfiability, and discovering new sub/super relations.

Our contribution is twofold. First, we aim at clarifying the semantics of part-whole relations by consid-ering the recent results of philosophical analyses on both mereological theories and foundational ontolo-gies. Second, we propose a new consistency check for automated reasoning services of description logics,which uses the introduced semantics for part-whole relations. We thereby aim at contributing to theoret-ical foundations of conceptual modelling as well as facilitating usage of part-whole relations in softwaresystems.

*Corresponding author: C. Maria Keet, KRDB Research Centre, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy. Email: keet@inf.unibz.it. Phone: +39 04710 16128, fax: +39 04710 16009

1570-5838/07/$17.00 c© 2007 – IOS Press and the authors. All rights reserved

2 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

To clarify the semantics of part-whole relations, this paper proposes a basic formal taxonomy of part-whole relations that includes both mereological and meronymic part-whole relations. The aim is to un-ambiguously specify the hierarchical relations between the various part-whole relations introduced in theliterature. The novelty of the approach taken is that the taxonomy is not merely a description of part-wholerelations tailored to one specific usage only, but it uses well-known foundational ontology aspects andan implementation-independent, formal characterisation, to enable portability across different conceptualmodelling languages and application scenarios. The obtained taxonomy and distinctions made will be jus-tified by formally defining them with First Order Logic (FOL) formulæ. To effectively use the part-wholerelations (and, in general, any hierarchy of relations), we propose a new reasoning service for automatedreasoners, which is called RBox compatibility. This new reasoning service ensures that a hierarchy of rela-tions is not only logically correct (as is currently the case with the extant reasoner software tools) but thatis also ontologically correct by checking a correspondence between the role hierarchy, the domain andrange restrictions for roles, and the class hierarchy. This enables earlier error detection and meaningfuluser feedback during conceptual model and domain ontology development.

In Section 2 we discuss related work on part-whole relations and disambiguate the types of part-wholerelations with a basic formal taxonomy in Section 3. Languages and tools are briefly considered in Section4, after which we proceed to automated reasoning over part-whole relations (and reasoning over hierar-chies of relations in general) in Section 5, where the new reasoning service for automated reasoners isintroduced. We close with conclusions and further research in Section 6.

2. Distinct Part-whole Relations: Related Works

Mereology is a sub-discipline in philosophy that concerns the formal ontological investigation of thepart-whole relation. Meronymy studies part-whole relations from a linguistics and cognitive science per-spective. There are meronymic relations that are not mereological. That is, there are usages of ‘part of’ innatural language and conceptual modelling that do not share the same properties as the part_of relation inmereology. The main semantic difference revolves around the (in)transitivity of the part-whole relation. Toappreciate the distinction, we first introduce some basic aspects of mereology (see Guizzardi (2005); Keet(2006b); Varzi (2004) for a more comprehensive introduction and overview) and subsequently summarisecontributions from meronymy and research into conceptual modelling from the perspective of engineeringusefulness. Based on this analysis, we propose a taxonomy of the various of part-whole relations in §3.

2.1. Ground Mereology

The lowest common denominator concerning theories of parthood is called Ground Mereology, whichstates that part_of is a relation capturing a partial order that is reflexive (1), antisymmetric (2), and transi-tive (3); all other versions share at least these constraints. This, however, does not mean (1-3) are uncon-tested; in particular transitivity of part_of receives attention, to which we return later.

∀x(part_of(x, x)) (1)

∀x, y((part_of(x, y) ∧ part_of(y, x)) → x = y) (2)

∀x, y, z((part_of(x, y) ∧ part_of(y, z)) → part_of(x, z)) (3)

Starting with these three basic formulæ that take part_of as a primitive relation, several other mereologicalpredicates can be built. A common one is the definition of proper part (4), which is asymmetric, irreflexiveand transitive. Note that in some mereological theories, the proper parthood is taken as primitive relation.

∀x, y(proper_part_of(x, y) , part_of(x, y) ∧ ¬part_of(y, x)) (4)

The relatively simple Ground Mereology is often extended with weak or strong supplementation, someversion of fusion, and atomicity, which we do not need for the current scope. Here, we focus on the as-

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 3

sertion that all mereological parthood relations are transitive, which is regularly discussed (e.g. Johans-son (2004); Odell (1998); Varzi (2006b)). Contrary to the straightforward mereological theories, exten-sions and modifications have been proposed to a) accommodate different types of part-whole relationsfor conceptual modelling and b) permit intransitivity in some cases of part-whole relations. But these twomotivations for usage of part-whole relations go against the mereological theories that claim there is justone, transitive parthood relation, which means something else is going on. We look into this issue in moredetail in the next Section.

2.2. Part-whole relations in meronymy and conceptual modelling

As was mentioned briefly in the previous Section, an important distinction exists between the mere-ological part_of relation and meronymic part-whole relations: the latter is not necessarily transitive andmay not fit within any mereological theory. That is, in our communication we use the term “part of”, butontologically one is not part of the other. For instance, Odell (1998)’s “member-bunch” part-whole rela-tion (subsequently referred to as member_of ) is a meronymic part-whole relation. For instance, Del Piero,an instance of Football player, is member of the Juventus football team, and he is also member of the JuventusTorino club, which in turn is member of the Italian football clubs federation Federcalcio, but Del Piero is nota member of Federcalcio. A different case of intransitivity is due to the mixed use of different part-wholerelations. Consider the canonical example of hand-musician-orchestra, where some instance of Musician ispart of some instance of Orchestra and a musician’s Hand is part of the instance of Musician, but it is falsethat that musician’s hand (or any other musician’s body part) is part of that orchestra. This intransitivityof ‘part-of’ with hand-musician-orchestra generalises also to the class-level, because no structural part(Hand) of a whole (Musician) propagates to be also a part of any collective whole (Orchestra); that is, herewe relate different kinds of things, which is the underlying cause of intransitivity of ‘part of’.

We now mention the efforts in the literature that have a list or a taxonomy of part-whole relations to sortout different usages of both the mereological parthood relation and meronymic part-whole relations. Thefirst proposal that introduced different types of part-whole relations was motivated by the linguistic use of‘part’, i.e. meronymy, and was made by Winston, Chaffin and Herrmann (WCH) (Winston et al. (1987)).Several successive articles analysed the WCH taxonomy and their modelling considerations (e.g. Artaleet al. (1996); Gerstl and Pribbenow (1995); Guizzardi (2005); Odell (1998)). For instance, Gerstl andPribbenow (1995) prefer a “common-sense theory of part-whole relations” instead, which is motivatedby differences in the compositional structure of the whole and they propose three different types: i) a ho-mogenous mass with quantities, ii) a collection of uniform elements, and iii) a complex of heterogeneouscomponents. They add a notion of “different views on the entities” in the sense that from one viewpointa complex may be just a collection, which can be a source of problems for semantic interoperability. Al-though they provide linguistic motivations by showing various examples supporting the existence of thesethree types of meronymic relations, they do not provide ontologically rigorous definitions for them. Sattler(1995) constrained her list of part-whole relations to types of “direct parthood”, which was motivated bythe constraint that Description Logic (DL) roles could not be transitive due to limitations on the language(ALC) and role hierarchies were not yet supported. The same problem motivated Schulz and Hahn (2000)to develop so-called SEP triples as engineering solution to circumvent the intransitive DL roles by rep-resenting the partonomy as a taxonomy1. Observe that both role hierarchies and role transitivity are sup-ported in the more recent expressive DLs SROIQ andDLRµ, as well as (ir)reflexivity and (a)symmetry,while reflexive antisymmetry is still an open issue (Horrocks et al. (2006)), thereby making them the twoformal knowledge representation languages known to be within the decidable fragment of FOL to supportmost of the aforementioned properties of parthood and proper parthood.

Odell (1998) proposes a list of six types of part-whole relations, providing descriptions and exam-ples. This is summarised in Table 1 and criticised by, among others, Guizzardi (2005); Keet (2006b);

1The three core items of a SEP triples are the Structure-concept node that subsumes one (anatomical) entity, called Entitynode, and the parts of that entity (the P-node). The is_a hierarchy is then built up by relating the P-node of a whole concept D tothe S-node of the part C, where in turn the P-node of C is linked to the S-node of C’s part.

4 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

Table 1Odell’s types of part-whole relations, in the first column, and our explanation in the second column.

Type of part-of Explanationcomponent – integral object Discrete type of part-whole relation, with atomsmaterial – object Constitution of objectsportion – object a) some amount of matter is part of the whole, and

b) scale-based partonomic relationsplace – area Where part-place cannot be separated from the whole-areamember – bunch Whole bunch is generally denoted with a collective noun and its members can

change over timemember – partnership Like member-bunch, but changing a member does destroy the whole

Motschnig-Pitrik and Kaasbøll (1999). Of these types, only component-integral object and place-areameet mereological parthood criteria, whereas the underspecified portion-object and member-bunch aremeronymic part-whole relations. Later, Motschnig-Pitrik and Kaasbøll (1999) reintroduced a small in-formal taxonomy that has all relations subsumed by a generic “meronymic relationship” that subsumes“(core) part-of”, which groups together mereological parthood with portion/mass, process/subprocessand place/area as well, then there is the member_of relation, a made_of for constitution, and a “noun-feature/activity” relation, which is better known in ontology under the name participates_in (see e.g. Ma-solo et al. (2003); Smith et al. (2005)). Opdahl et al. (2001); Barbier et al. (2003) distinguish betweenthree kinds of properties of “whole-part relationships” – “primary” (asymmetry, antisymmetry, and emer-gent property), “secondary” (transitivity or intransitivity, shareability, mutability, and separability), and“consequent” (ownership, propagation of operations, and abstraction)2 – and provide a characterisationin UML’s Object Constraint Language (OCL). The differentiation of kinds of properties exhibits manycurious features; we discuss two. First, we mention the prioritization of, for conceptual data modelling,more and less important requirements to represent part-whole relations, which blurs the distinction be-tween requirements for one object-oriented language and properties of the part-whole relation. The au-thors intermingle object-oriented application-driven modelling considerations – such as software objectcreation & destruction, encapsulation, and ownership – and the (non-)representation of attributes with theBunge Wand Weber (BWW) “ontology used as a model” the authors had taken originally. BWW doesmake a distinction between existence of properties in reality and their representations (see also Guizzardi(2005)). Put differently, for Opdahl et al. (2001) (p393), a part-whole relation should not be represented ina UML class diagram if some of the classes’ “resultant and emergent” attributes are not represented, eventhough the part-whole relation exists in reality between the affected classes. Second, primary properties ofpart-whole relations do not correspond to combinations of properties of the part_of and proper_part_ofrelations in mereology, thereby setting aside a great deal of ontological investigation and usable results(see Section 3). Moreover, eventually they have removed transitivity from the list of primary properties ofpart-whole relations —for conceptual data modelling—, based on a single confused example about intran-sitivity between body parts, person and research group. The weakness of their argument can also be tracedback to the problem of fancying to represent some class attributes in one’s UML class diagram, or not,versus properties of universals and their relations, and of mixing different types of part-whole relationsand relata. Representing part-whole relations for actual domains indeed is not an easy task, but instead ofavoiding representational issues and “thereby confirm” (Opdahl et al. (2001)) demotion of transitivity tosecondary property, one should try to resolve the problematic aspects.

In contrast to these earlier attempts, Guizzardi (2005) provides criteria for two types of part-whole re-lations and distinguishes between the mereological part_of relation and three other part-whole relations,being sub_quantity_of to relate amounts of matter, sub_collection_of that actually represents a set-subsetrelation (e.g., a group of seminar attendees where there is a subset of the attendees with nationality Dutch),

2“a primary characteristic is a necessary condition for a relationship in an OO-model to be a WP relationship.” Secondary prop-erties do not hold for all OO-model whole-part relations, and a consequential characteristic is “a logical (or natural) consequenceof one or more of the primary ones” (Opdahl et al. (2001)).

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 5

Part-whole relation

mpart_of ((Meronymic) part-whole relation)

part_of (Mereological part-of relation)

member-of constitutes sub-quantity-of participates-in involved-in spatial-part-of

f-part-of

s-part-of

located-in contained-in member-of’

… … … …

… …

Fig. 1. Taxonomy of basic mereological and meronymic part-of relations. s-part-of = structural part-of; f-part-of = functionalpart-of. Dashed lines indicate that the subtype has additional constraints on the participation of the entity types; ellipses indicateseveral possible finer-grained extensions to the basic part-whole relations.

and member_of. Although Guizzardi (2005) describes clearly for each type of relation the graphical UMLnotation and their formal properties, he underspecifies the stereotypes (or categories) of the entity typesinvolved in a part-whole relation and omits both parthood between processes and mereotopological rela-tions (parts and connection/space).

How to figure out differences between parthood and parthood-like relations based on a FOL formal-ization, and how to structure the proposed part-whole relations in a clear manner, is the topic of the nextSection. In particular, it will become clear that in case of different types of part-whole relations, differ-ent categories of entity types (universals) will be related. Provided one makes these required distinctions,transitivity still holds for parthood and can be easily distinguished from intransitive part-whole relations.

3. A Formal Taxonomy of Part-Whole Relations

To provide an unambiguous way of dealing with part-whole relations in conceptual data models (anddomain ontologies), we propose a formal taxonomy of part-whole relations. The taxonomy structures thevarious part-whole relations, as introduced and/or discussed by e.g. Artale et al. (1996); Barbier et al.(2003); Bittner and Donnelly (2005); Gerstl and Pribbenow (1995); Guizzardi (2005); Johansson (2004);Keet (2006a); Motschnig-Pitrik and Kaasbøll (1999); Odell (1998); Sattler (1995); Schulz et al. (2006);Rector et al. (2006); Smith et al. (2005); Tan et al. (2003); Vieu and Aurnague (2005); Winston et al.(1987), by taking into account ontological distinctions3. The proposed taxonomy is depicted in Fig.1and is explained in the remainder of this Section, including formal definitions for the leaf types of part-whole relations. The taxonomy is a balance between typing ontologically-motivated relations useful forconceptual modelling up to the minimum level of distinctions to gain benefit from specifying part-wholerelations more precisely, yet avoiding ontological exuberance that would deter conceptual modellers fromusing it in practice during the conceptual analysis stage; where applicable, current cut-off points will bejustified in the explanation below.

3.1. Overview and preliminaries

The first principal distinction in the taxonomy is made between transitive and intransitive part-wholerelations. The prime reason why this ontological distinction exists has to do with transitivity of the mere-ological parthood relation versus other part-whole relations. Successive distinctions between the relations

3The taxonomy does not deal with other facets of parthood relations, such as intra-part relations, the inverse relation has_part,and if the parts together are all parts that make up the whole; these aspects are beyond the scope of this article. For a discussionand various options to address such issues, see e.g. Barbier et al. (2003); Guizzardi (2005); Lambrix and Padgham (2000);Motschnig-Pitrik and Kaasbøll (1999); Opdahl et al. (2001).

6 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

PTParticular

EDEndurant

PDPerdurant

PEDPhysicalEndurant

NPEDNon-physical

Endurant

ASArbitrary

Sum

EVEvent

STStative

ACHAchievement

ACCAccomplishment

STState

PROProcess

NPOBNon-physical

object

MOBMental object

SOBSocial object

POBPhysicalobject

FFeature

MAmountof matter

NAPONon-agentive

physical object

APOAgentive

physical object

…

…

… … … …

… …

Fig. 2. Graphical rendering of a section of the foundational ontology DOLCE. In colloquial natural language communication,endurant roughly maps to entity types and perdurant to processes or (objectified) relations/associations in conceptual models.

are made based on the categories of the entity types participating in the relation—also called relata, domainand range restriction, stereotypes of object types. The types of part-whole relations are disjoint. Furtherdistinctions can be made on finer-grained categories of participating entity types and on other propertiesof the relation, such as existential dependence of the part on the whole or vice versa.

To be able to talk about the categories of the entity types involved in part-whole relations, we haveto consider foundational ontologies from which we can borrow several top-level categories. Of the sev-eral extant foundational ontologies, such as DOLCE (Masolo et al. (2003)), BFO4, OCHRE (Schneider(2003)), SUMO, and GFO (Herre and Heller (2006)), we chose DOLCE, the Descriptive Ontology forLinguistic and Cognitive Engineering, because it is the most comprehensively formalised one, has a map-ping to OWL5, and is used across several subject domains for development of ontology-driven informationsystems6. DOLCE includes the afore-mentioned Ground Mereology (as it uses Atomic General Exten-sional Mereology for atemporal parthood), and the definitions introduced in this paper are fully compatiblewith DOLCE’s formal characterisation. A portion of the DOLCE foundational ontology that is relevantfor the purposes of this paper is depicted in Fig.2 (directly subsumed universals are disjoint (Masolo etal., 2003)). In the remainder of this section, we will use the abbreviated names of Fig.2, such as ED forendurant and POB for physical object, after their first introduction in the text.

To avoid overloading terms, the non-mereological part-whole relation is labelled with mpart_of . Thispart-whole relation is included in the taxonomy mainly for structuring purposes and is not intended forgeneral use. Observe also that mpart_of is ‘non-transitive’, that is, neither transitive nor intransitive7,because it is not the case that intransitivity holds for all its subsuming part-whole relations with all theirrelated instances. Thus, its semantics differs from the mereological part_of , which is assumed transitive.Because of this distinction, the top relation in the taxonomy, part-whole-relation, is necessarily alsonon-transitive. Both part_of and mpart_of inherit from part-whole-relation the typing of the relata(domain & range restriction), assumed to be the DOLCE’s top-category Particular (PT ). For explanatory

4http://www.ifomis.uni-saarland.de/bfo/home.php5Web Ontology Language, which is based on description logics. http://www.w3.org/TR/owl-features/.6See for an overview: http://www.loa-cnr.it/DOLCE.html.7non-transitive is not a new property but a short-hand notation for absence of declaring transitivity or intransitivity, which we

use when it is known that the relation is neither transitive for all cases nor intransitive

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 7

purposes only, subsequent definitions in the meronymic branch contain the subscript “it” to denote anintransitive relation (∀x, y, z(R(x, y) ∧R(y, z) → ¬R(x, z))) or “n” for non-transitive.

3.2. Leaf types of part-whole relations

Here we describe the meronymic leaf types first and the mereological parthood relations afterward.The member_of relation (in the literature also called member-bunch, collection, collective or aggrega-

tion) belongs to the meronymic branch. The whole-side of the relation is generally denoted with a collec-tive noun, i.e. a social entity with aggregations like Herd or Orchestra. The part-side are physical entities orthe role they play, such as Sheep and Musician, respectively. Given that roles are dependent on the bearer,a physical object in its turn, we formalise member_of with the following definition.

∀x, y(member_ofn(x, y) , mpart_of(x, y) ∧ (POB(x) ∨ SOB(x)) ∧ SOB(y)) (5)

This member_of constrains the ‘part’ side of the relata to either physical objects (POB) or their rolesas non-physical social objects (SOB) (a simplification of the work by Masolo et al. (2004)), while thewhole the parts are aggregated into are constrained to be social objects SOB. With (5), it is easy torepresent Odell (1998)’s “member-partnership” example relation for Husband and Wife in a partnershipMarriage. The only addition is that the whole is existentially dependent on the part and vice versa: let εbe existential dependence (see also Guizzardi (2005)), then we can add a subtype member_of ′ which has“ε(x, y) ∧ ε(y, x)” added to definition (5).

The material-object relation, categorised as part-whole relation by Winston et al. (1987); Sattler (1995);Gerstl and Pribbenow (1995); Odell (1998), corresponds ontologically to constitution where a POB isrelated to an amount of matter (M ) it is made of (see section 3.3.3 in Masolo et al., 2003, for detail andjustification)—e.g. Statue and the Marble it is constituted of. Amounts of matter are generally denoted withmass nouns that are not countable. In natural language, the inverse constituted_of is used more often,where the whole is constituted of its material-parts.

∀x, y(constitutesit(x, y) ≡ constituted_ofit(y, x) , mpart_of(x, y) ∧ POB(y) ∧M(x)) (6)

The sub-quantity-of relation, also called quantity-mass or portion-object (e.g. Sattler (1995); Odell (1998);Guizzardi (2005)), relates a smaller part-amount of matter (M ) to a whole-matter (M ), where the amountsof matter are either the same type of stuff, e.g., a glass of Wine—in this case we require an additional mea-sure for its quantity like glass & bottle or millilitre & litre—or, similar to Guizzardi (2005), the part-Mis a different type of matter than the whole-M—e.g. sub_quantity_of(Salt, SeaWater). These examples areontologically distinct, giving rise, in the first case, to a transitive sub_quantity_of ′ relation and to anintransitive sub_quantity_of ′′ relation in the second case. In addition, in the second case, a part-M mayhave undergone a chemical reaction in the whole-M and, strictly speaking, not exist anymore comparedto the part-M in isolation or existing in a different state (e.g., the molecules have released a hydrogenatom upon dissolving in the whole-M ). Further, specific quantities may not matter for representing basicknowledge in ontologies, such as representing sub_quantity_of(Alcohol, Wine), but this is needed for concep-tual data models where recording data on percentages of alcohol in beverages is needed. Therefore, (7)has the lowest common denominator, but it can benefit from further disambiguation.

∀x, y(sub_quantity_ofn(x, y) , mpart_of(x, y) ∧M(x) ∧M(y)) (7)

The last meronymic relation is participates_in, a noun-feature/activity in linguistics (Motschnig-Pitrikand Kaasbøll (1999)), which relates an entity of the category endurant (ED) to the process (perdurantPD) it participates in (Masolo et al. (2003))—e.g. an Enzyme that participates in a CatalyticReaction.

∀x, y(participates_init(x, y) , mpart_of(x, y) ∧ ED(x) ∧ PD(y)) (8)

From this characterisation, it is clear why meronymic part-whole relations are at least non-transitive, butgenerally intransitive: the category of the part is usually different from the category of the whole8, there-

8The only exception being sub_quantity_of , which is under-specified.

8 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

fore one can neither make a chain with the same relation nor concatenate them with the mereologicalrelations and assume transitivity holds.

We now consider the mereology branch of the taxonomy, where all types of relation are transitive andthey can be formalised (specialised further) with proper parthood (4) as well.

We first encounter the involved_in relation, which relates two perdurants (9); for instance, Chewingis involved in Eating. This definition concurs with the parthood argument restriction “(Ad2)” in DOLCE,P (x, y) → (PD(x) ↔ PD(y)), and hereby is rendered more accessible for conceptual modelling.

∀x, y(involved_in(x, y) , part_of(x, y) ∧ PD(x) ∧ PD(y)) (9)

We distinguish between two mereotopological relations for endurants (mereological relations that takeinto account space or location), one according to a 3-dimensional containment (10) and another for 2-dimensional location (11). The definitions refer to both the endurant itself and the region (R) it occupies.DOLCE has an elaborate formalisation to refer to the region of an endurant (using quality, quale, andregions), which is abbreviated here with the properties has_3D and has_2D, respectively, because thisattribute-approach generally provides sufficient details in conceptual models for applications9. One can doaway with the 2D/3D distinction and just refer to any kind of spatial region, but the distinction is includedbecause it is a recurring relation in the informal discussions about part-whole relations for conceptualmodelling (e.g. Gerstl and Pribbenow (1995); Keet (2006a); Motschnig-Pitrik and Kaasbøll (1999); Odell(1998); Winston et al. (1987)) and can be accommodated for easily.

∀x, y(contained_in(x, y) , part_of(x, y) ∧R(x) ∧R(y)∧∃z, w(has_3D(z, x) ∧ has_3D(w, y) ∧ ED(z) ∧ ED(w)))

(10)

∀x, y(located_in(x, y) , part_of(x, y) ∧R(x) ∧R(y)∧∃z, w(has_2D(z, x) ∧ has_2D(w, y) ∧ ED(z) ∧ ED(w)))

(11)

For instance, contained_in(John’s address book, John’s bag). Containment is not always conceptualised as aparticular type of parthood (Bittner and Donnelly (2005); Odell (1998); Schulz et al. (2006)), but, first,contained_in always meets the above-mentioned mereological parthood and proper parthood proper-ties. Second, those considerations on conceptualizing parts & space exhibit a hopping back and forth be-tween considering entity only, entity & region it occupies, and region only. With examples such as con-tained_in(actin filament, cell) and many others in biology, both parthood and containment hold, which meansthat x & z and y & w coincide exactly. To address these subtle, but important, distinctions, we explicitly in-clude structural parthood (see below) in the taxonomy as different from containment. The same argumentholds for location. Examples for location are located_in(Amsterdam, North Holland) or located_in(Mont Blanc,Alps). Observe that the examples for location permit a finer-grained specification regarding the relata: theformer relates city-province, which are entities by social convention, whereas the latter relates two physi-cal entities (mountain and mountain range). Such ontological exuberance is not included here, because, itis generally less relevant for conceptual modelling and software system development in practice and doesnot result in fallacious transitivity.

Last, we constrain the structural parthood relation (12).

∀x, y(s_part_of(x, y) , part_of(x, y) ∧ ED(x) ∧ ED(y)) (12)

We can further constrain (12) by defining two subtypes of s_part_of for POBs or to relate two NPOBsto ensure POBs and NPOBs are not interleaved (13, 14).

∀x, y(s_part_of ′(x, y) , part_of(x, y) ∧ POB(x) ∧ POB(y)) (13)

∀x, y(s_part_of ′′(x, y) , part_of(x, y) ∧NPOB(x) ∧NPOB(y)) (14)

9Note that the relations do not reflect the full range of mereotopological and mereogeometrical complexities either (Borgoand Masolo (2007); Varzi (2006a)), some of which could be useful for geographic and biological information systems, but theyare less relevant for the more common enterprise domain modelling. It is a point of further research how these first order logictheories can be transformed and rendered usable with formal conceptual modelling languages.

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 9

In addition, and analogous to the member_of ′, we can add a constraint to the part-side of s_part_ofto make it a functional part (Vieu and Aurnague (2005); Guizzardi (2005)) and label it f_part_of ; in-formally, individual functional dependence captures that for x to function as X then y must function asY . Functional parthood is, however, motivated by—as opposed to ‘due to’—linguistics and has not (yet)been identified explicitly as an important part-whole relation for conceptual modelling, although mod-ellers surely have dealt with representing functional parthood. For instance, that f_part_of(Car Engine, Car)holds, with the meaning that the entity type Car denotes the set of canonical cars and each car cannot func-tion normally without its canonical, working, engine (that instantiates the entity type Car Engine). It hasbeen included to demonstrate ease of extension of the taxonomy if the conceptual modelling communitydesires further specialisation with other properties than the core mereological properties of parthood. Thisconcludes the characterisation of the eight leaf types.

Last, observe that by adhering to Ground Mereology in the taxonomy of part-whole relations, we get sixother mereological relations ‘for free’. These relations (15-20) are generally useful for conceptual mod-elling, but in particular for conceptual modelling for Geographical Information Systems and applicationsfor biology and biomedicine; they are as follows (after Varzi (2004)):

overlap(x, y) , ∃z(part_of(z, x) ∧ part_of(z, y)) (15)

underlap(x, y) , ∃z(part_of(x, z) ∧ part_of(y, z)) (16)

overcross(x, y) , overlap(x, y) ∧ ¬part_of(x, y) (17)

undercross(x, y) , underlap(x, y) ∧ ¬part_of(y, x) (18)

proper_overlap(x, y) , overcross(x, y) ∧ overcross(y, x) (19)

proper_underlap(x, y) , undercross(x, y) ∧ undercross(y, x) (20)

4. Using Part-Whole Relations: Languages and Tools

In this Section, we assess the options provided by several modelling languages for representing part-whole relations and integrating a part-whole taxonomy, and to use them for the conceptual modellingor domain ontology development stage. Conceptual data models and ontologies are not synonymous,primarily differing in what part of reality or knowledge for some subject domain is represented, but therequirements for dealing with representing a taxonomy of relations is the same for their languages andtools. That is, requirements for how to represent part-whole relations and the taxonomy in the languageand the basic functionality it requires from the CASE and ontology development tools are identical for thetwo. Based on the semantics of part-whole relations as presented in the previous Section, we can list thefollowing four requirements that every language intending to represent part-whole relations should pass:

1. Represent at least Ground Mereology,2. Express ontological categories and their taxonomic relations,3. Having the option to represent transitive and intransitive relations, and4. Specify the domain and range restrictions (/relata/entity types) for the classes participating in a

relation.Several engineering solutions are possible to implement the basic taxonomy of part-whole relations, butthe available options depend on the conceptual modelling or ontology language. In the following, wefocus on UML, EER, and Object-Role Modeling (ORM) as conceptual data modelling languages, and onDescription Logics (DLs) as dual-purpose conceptual modelling and ontology languages.

EER, ORM, and DLs do not have special constructors to represent the part-whole relation, althoughsome are in favour of giving it a first-class citizen status (Artale et al. (1996); Bittner and Donnelly (2005);Keet (2006a); Lambrix and Padgham (2000); Sattler (1995); Shanks et al. (2004)). UML, on the otherhand, implements two versions of the part_of relation: composite and shared aggregations (Object Man-

10 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

«ED»CarChassis

«ED»Car

«PD»Chewing

«PD»Eating

A. B.

part_of involved_in

C.

CarChassis part of Car.Each CarChassis part of exactly one Car.For each Car, exactly one CarChassis part of that Car.

ED(ID)

CarChassis Carpart of

[part_of]

Fig. 3. Two examples of stereotyping with DOLCE categories in a UML class diagram using (A) composite and (B) sharedaggregation. Without stereotyping as in ORM2 (C), the object types are subsumed by the DOLCE category ED (note that“[part_of]” is the role name and “part of” the reading label for the verbalization).

agement Group (2005)). The difference between the two UML aggregation relations is that with compos-ite aggregations, the part is existentially dependent on exactly one whole, whereas for shared aggregationthere is no constraint on the multiplicity. However, their “precise semantics [...] varies by application areaand modeler” (Object Management Group (2005)), and presumably thus could be used for any of the typesof part-whole relations described in Fig.1. Possibilities to include different types of part-whole relationsin UML are: modifying the meta-model, adding new icons, defining stereotypes for the categories of thedomain and range restrictions, and OCL constraints to describe the properties of the part-whole relation(Barbier et al. (2003); Guizzardi (2005); Motschnig-Pitrik and Kaasbøll (1999); Tan et al. (2003)); a sim-ple example is shown in Fig.3.A-B. The relevant DOLCE categories are mapped straightforwardly onto astereotype for each category and ensured that they form a taxonomy. Adding different icons for the 8 leaftypes of part-whole relations is not advisable, because it would result in cluttered diagrams. Instead, wepropose that, upon adding an association, one can select the appropriate part-whole relation from a pre-defined list. An additional possibility for ORM tools is to use the names of the relations in the fact editorwhere the user writes pseudo-natural language sentences to verbalize the fact type. This can be both tosupport the user in selecting a particular part-whole relation and, given an entered string for the relation, tocheck the types of object types surrounding the label of the part-whole relation. For subrelations, the ORMconstraint of subset over binary relations is a restricted option, or it can be added through subtyping objec-tified (nested) relations. Given that ORM does not use stereotypes, one can add the relations to the ORMmetamodel or, as depicted in Fig.3.C, have the object types subsumed by the DOLCE category-as-objecttype.

DLs, SHIQ and SROIQ10 (Horrocks et al. (2006)) and DLRs (Calvanese and De Giacomo (2003))in particular, have the ability to structure DL-roles into hierarchies (called the Role Box, or RBox), ex-press both domain & range restriction and cardinality constraints on the participation of entities, representa concept hierarchy (class taxonomy) with Terminological Axioms (TBox), and can support most of themereological parthood properties (see Table 2 for details). Finally, DLs have tools, such as iCOM, forconceptual modelling development (Franconi and Ng (2000)) and “CASE”-like tools for ontology devel-opment, such as Protégé, both of which are linked to a DL reasoner such as Racer or Pellet, which enablesthe modeller to easily specify both a DL-concept hierarchy and a DL-role hierarchy (“object properties”in Protégé) and run the reasoner over it with one mouse-click. The next Section discusses how to integratethe two hierachies and how to reason over them.

10SROIQ extended with data types is a basis for OWL 1.1, with syntax at: http://www-db.research.bell-labs.com/user/pfps/owl/overview.html (Editor’s draft 14-6-2006).

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 11

Table 2Properties of parthood and proper parthood compared to their support in DLRµ (with fixpoint extension (Calvanese et al.(1999))), SHOIN and SROIQ. ∗: properties of the parthood relation (in Ground Mereology); ‡: properties of the properparthood relation (in Ground Mereology).

Feature ⇒ Reflexivity ∗ Antisymmetry ∗ Transitivity ∗ ‡ Asymmetry ‡ Irreflexivity ‡

Language ⇓DLRµ + - + + +SHOIN - - + + -SROIQ + - + + +

5. Reasoning over a Part-Whole Taxonomy

Although the features in the current conceptual modelling languages and CASE tools are somewhatlimited in fully supporting the representation of the part-whole taxonomy together with the relations’ prop-erties, it is possible to at least specify the domain and range for the relations. The natural next step is toensure that this typing is done and used correctly, therefore we now focus on automated reasoning and weshow that managing a taxonomy of relations requires a new reasoning service to make ontologically cor-rect inferences over both conceptual models and domain ontologies. The aim of such a reasoning serviceis to help a modeller to be more precise in representing part-whole relations and to achieve an effective useof the part-whole taxonomy. A decision procedure to guide the modeller was proposed by Keet (2006a),for which several implementation options were suggested: a cheat-sheet, drop-down box in a CASE toolto type the relation, or software-support for the decision procedure with questions and examples corre-sponding to each decision point. However, none of the suggested options can compute correctness, or atleast logical consistency, of the modelling decisions taken by the modeller about the relations. Given thatrepresenting part-whole relations is not an easy task, it is highly desirable to offer automated reasoningsupport to aid the modeller in checking consistency and satisfiability of the conceptual model and to derivenew information entailed in the model.

We choose Description Logics to formulate the desired reasoning service. DLs have been shown usefulfor reasoning both over conceptual models like EER, ORM, and UML (Artale et al. (2003); Baader et al.(2002); Berardi et al. (2005); Calvanese et al. (1998, 1999); Keet (2007)) and ontology languages suchas OWL-DL, OWL-Lite11, OWL 1.1, and DL-Lite (Calvanese et al. (2005)). At present, most propertiesof the mereological theories can be represented in expressive DLs (Calvanese and De Giacomo (2003);Horrocks et al. (2006)), except for antisymmetry; see Table 2 for details.

For the current purpose, we are mainly interested in reasoning over roles given a role hierarchy (rep-resented by the Role Box, RBox) and a DL-concept hierarchy (represented by the Terminological Box,TBox). For our purposes, it is enough to introduce briefly the simple ALCI DL language, which has al-ready sufficient expressivity to support the reasoning service we are interested in.ALCI is a sub-languageof both the proposed OWL 1.1, which is SROIQ with datatypes, and the DLR family of DL languages(Calvanese and De Giacomo (2003)), which were specifically developed to provide a formal underpinningand unifying paradigm for conceptual modelling languages and permit automated reasoning over concep-tual models. With respect to the formal apparatus, we will strictly follow the concept language formalismpresented in Baader et al. (2002). Basic types of ALCI are concepts and roles. A concept—sensu DL—isa description gathering the common properties among a collection of individuals; from a logical pointof view, it is a unary predicate ranging over the domain of individuals. Inter-relationships between theseindividuals are represented by means of roles, which are interpreted as binary relations over the domainof individuals. According to the syntax rules of Fig. 4, ALCI concepts (denoted by the letters C and D)are built out of atomic concepts (denoted by the letter A) and atomic roles (denoted by the letter P ). In thefollowing we use ∃R as a shortcut for ∃R.>. As usual, an ALCI interpretation is a pair, I = (∆I , ·I),where ∆I is a non-empty set of objects (the domain of I) and ·I an interpretation function such that, for

11http://www.w3.org/TR/2004/REC-owl-semantics-20040210/syntax.html.

12 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

C, D → A | (atomic concept)> | (top)⊥ | (bottom)¬C | (complement)C uD | (conjunction)C tD | (disjunction)∀R.C | (univ. quantifier)∃R.C | (exist. quantifier)

R, S → P | (atomic role)R− (inverse role)

AI ⊆ ∆I

>I = ∆I

⊥I = ∅(¬C)I = ∆I \ CI

(C uD)I = CI ∩DI

(C tD)I = CI ∪DI

(∀R.C)I = {a ∈ ∆I | ∀b.RI(a, b) ⇒ CI(b)}(∃R.C)I = {a ∈ ∆I | ∃b.RI(a, b) ∧ CI(b)}

P I ⊆ ∆I ×∆I

R−I = {(a, b) ∈ ∆I ×∆I | (b, a) ∈ RI}

Fig. 4. Syntax and Semantics for the ALCI Description Logic

every concept C, and every role R, we have CI ⊆ ∆I and RI ⊆ ∆I ×∆I . Generic concepts and rolesare interpreted by I according to the semantic equations of Fig.4.

A knowledge base in this context is a pair Σ = (T ,R) where T is a set of terminological axioms(TBox) of the form C v D (general concept inclusion axiom), and R is a set of role axioms (RBox) ofthe form R v S (subrole axiom) and R v C1 × C2 (Domain & Range axiom)12. An interpretation Isatisfies C v D iff CI ⊆ DI , R v S iff RI ⊆ SI , and R v C1 × C2 iff RI ⊆ CI

1 × CI2 . A knowledge

base Σ is satisfiable if there is an interpretation I which satisfies every axiom in Σ; in this case I is calleda model of Σ. Σ logically implies an axiom α (written Σ |= α) if α is satisfied by every model of Σ.A concept C (role R) is satisfiable, given a knowledge base Σ, if there exists a model I of Σ such thatCI 6= ∅ (RI 6= ∅×∅), i.e. Σ 6|= C v ⊥ (Σ 6|= ∃R v ⊥), thus, role satisfiability can be reduced to conceptsatisfiability). We illustrate the DL syntax in the following example.

Example. The ORM example in Fig.3 can be represented in ALCI with the following knowledge base,where we use [part_of] as an atomic DL role, omit the uniqueness and object type reference modes asthey are not relevant for the current purpose, and ED is endurant, PD perdurant, and PT particular (seeFig.2):CarChassis v ED

Car v ED

CarChassis v ∃PARTOF.CarCar v ∃PARTOF−.CarChassis

Further, we can describe, e.g., the domain and range restriction of PARTOF, one of its subtypesINVOLVEDIN, and its domain and range restriction:PARTOF v PT × PT

INVOLVEDIN v PARTOF

INVOLVEDIN v PD × PD

The DOLCE categories ED and PD are both subconcepts of PT and are disjoint:ED v PT

PD v PT

ED v ¬PDGiven this knowledge base, it is clear that it is logically correct that the car chassis is PARTOF of the carand that it can never be INVOLVEDIN the car in the current state of the knowledge base. Indeed, both carand car chassis are subconcepts of ED, which in turn is a subconcept of PT, hence, PARTOF can be used.Furthermore, INVOLVEDIN is typed with PD, i.e., it can be used to relate perdurants only, but we have thatED and PD are disjoint, and car chassis and car are both types of ED, hence, declaring INVOLVEDIN for thecar and car chassis would lead to a logical inconsistency. ♦

12Note that Domain & Range axioms are a shortcut for the following two axioms: ∃R v C1, ∃R− v C2, with C1, C2 genericconcepts.

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 13

Reasoning over a role taxonomy means checking for role satisfiability and checking the compatibilityof domain & range axioms with respect to subrole relations holding in the RBox. Role consistency is anissue when dealing with both RBoxes and TBoxes, since an unsatisfiable concept can be associated toeither the domain or the range of a role leading to an inconsistent role. Extant reasoners, such as Pellet,FaCT, and Racer, can check role satisfiability by reducing it to concept satisfiability—i.e. by checkingwhether Σ 6|= ∃R v ⊥. In this paper we introduce a new reasoning service called RBox compatibilitywhere the domain and range restrictions for roles are checked against the RBox and the TBox. We showthat this new service can help in avoiding unwanted logical consequences of an RBox over a set of con-cept hierarchies expressed in the TBox. We demonstrate with an example both the relevance of reasoningover an RBox and the problem with extant reasoning services that do not deal adequately—in the senseof deriving ontologically correct results—with a role hierarchy. We thus provide a formal definition forthe RBox compatibility test and then show how this new reasoning service can help in avoiding ontolog-ically undesired consequences. For the example, the ontology development tool Protégé v3.2 beta is used(together with the Racer reasoner) because the screenshots are more illustrative than those of the formalconceptual modelling tools and the issue is the same for both domain ontologies and formal conceptualmodels.

Example (cont’d). Take the three top-most categories of DOLCE (as in Fig.2) and add them as classes inProtégé. Then add a role hierarchy with pwrelation at the top that subsumes part-of that in turn subsumesinvolved-in with their proper domain and range restrictions, as defined in Section 3. Then, relate Chewingto Eating with the involved-in relation, and Chassis to Car through the part-of relation. This is depicted inFig.5.A1-B as screenshots from Protégé v3.2 beta. For illustrative purpose, we have created another classhierarchy too (Fig.5.A2) where Chassis and Car are not subconcepts of endurant ED anymore, but of par-ticular PT instead. Further, we have another scenario in Fig.5.C with an “incompatible” role hierarchy, i.e.a role hierarchy where the domain and range restrictions are inverted compared to the correct scenario,since now part-of’s subrole involved-in can be used to relate anything to anything whereas part-of itself canonly relate perdurants.

Using the reasoning options of Protégé with Racer, we obtain the results as summarised in Fig.6. Choos-ing the TBox (A1) with the “correct” RBox (B) shows, as expected, that the ontology is fine. Testing theTBox (A1) with the “incompatible” RBox (C), it says Chassis is inconsistent, whereas using the TBox(A2) with the “incompatible” RBox (C) reclassifies Chassis as a type of perdurant. Although with relationto the scenarios using the RBox (C), these deductions are logically correct, we claim here that the reasonerought to have found a compatibility issue in the RBox (C), because the domain and range restrictions of asubrole cannot be more general (higher up in the TBox) than those of its parent role. ♦

We are now able to define formally the new proposed reasoning service that we call RBox compatibil-ity. We start first with some notation. In the following Definition we assume that for each role there isprovided exactly one Domain & Range axiom. Without loss of generality, we can assume that the axiomR v >×> holds when an explicit Domain & Range axiom is lacking.

Definition 1 (User-defined Domain and Range Concepts). Let R be a role and R v C1×C2 its associ-ated Domain & Range axiom. Then, with the symbol DR we indicate the User-defined Domain of R—i.e.,DR = C1—while with the symbol RR we indicate the User-defined Range of R—i.e., RR = C2.

We now define the new reasoning service, RBox compatibility, that checks the compatibility of Domain& Range axioms with respect to both the role hierarchy holding in the RBox and the concept hierarchyholding in the TBox. The tests of the RBox compatibility service are not only necessary but also sufficientfor finding domain-range problems, because it covers each permutation of domain and range of the parentand child relation in the role hierarchy.

Definition 2 (RBox Compatibility). For each pair of roles, R,S, such that 〈T ,R〉 |= R v S, checkwhether:

14 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

A1. Class hierarchy with asserted conditions

B. Correct role box (object properties) C. Wrong role box (object properties)

A2. Other class hierarchy with

the same asserted

conditions

Fig. 5. Two class hierarchies and two role (object property) hierarchies in Protégé.

3. A1+C+racer: class hierarchy is inconsistent 4. A2+C+racer: Chassis reclassified as PD

1. A1+B+racer: ontology OK 2. A2+B+racer: ontology OK

5: Required inference result A1/A2+C+reasoner:

role hierarchy is inconsistent, with inconsistent roles “domain & range involved-in and part-of are inconsistent”, which can be fixed by the user, else the reasoner suggests:

Computing superroles reasoner log: “involved-in Moved to pwrelation“ and “part-of Moved to involved-in”

Fig. 6. The examined four combinations of automated reasoning over the in Fig.5 shown class hierarchies and object property(Role Box) hierarchies with the current results of the reasoner (Racer) when checking consistency and computing the inferredhierarchy (classifying the taxonomy).

Test 1. 〈T ,R〉 |= DR v DS and 〈T ,R〉 |= RR v RS;Test 2. 〈T ,R〉 6|= DS v DR;Test 3. 〈T ,R〉 6|= RS v RR.

An RBox is said to be compatible iff Test 1 and (2 or 3) hold for all pairs of role-subrole in the RBox.

A formal conceptual model or domain ontology that does not respect the RBox compatibility criterioncan be considered as ontologically flawed. Checking for RBox compatibility and, thus, for ontologicalRBox correctness, can be done by using the classical (for DL reasoners) subsumption reasoning service.In particular, we propose the following actions whenever the above defined tests fail. If Test 1 does nothold, we raise a warning that domain & range restrictions of either R or S are in conflict with the rolehierarchy proposing either

(i) To change the role hierarchy or(ii) To change domain & range restrictions or

(iii) If the test on the domains fails, then propose a new axiom R v D′R×RR, where D′

R ≡ DRuDS13,

13The axiom C1 ≡ C2 is a shortcut for the axioms: C1 v C2 and C2 v C1.

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 15

which subsequently has to go through the RBox compatibility service (and similarly when Test1 fails on range restrictions).

If Test 2 and Test 3 fail, we raise a warning that R cannot be a proper subrole of S but that the tworoles can be equivalent. We then propose two actions: either

(a) Accept the possible equivalence between the two roles or(b) Change domain & range restrictions.

Please note that the above actions should allow the user to leave unchanged both the TBox and the RBox,since both tests do not imply any real logical inconsistency—i.e., the proposed actions are not mandatory.Thus, our proposal refines the notion of correctness by distinguishing between the notion of logical cor-rectness and that of ontological correctness. We demonstrate the application of the RBox compatibilityservice with the running example.

Example (cont’d). Observe that both Protégé deductions—i.e., Chassis inconsistent in scenario A1+C,and Chassis as a perdurant in scenario A2+C—are just logical consequences of disregarding the ontolog-ical incorrect (but logically consistent) assumptions contained in the RBox (C). Put differently: with theRBox compatibility service in place, one obtains these deductions by ignoring all warnings raised andsuggestions proposed by the service.

Now consider RBox (C) with the RBox compatibility service, then Test 1 fails because both the do-main and range of involved-in, particulars PT, are parent concepts of its superrole (part-of) domain and rangerestrictions that are set to perdurants PD. Between the possible options (i)-(iii) that an user can choose,the second one is the most appropriate. In particular, the user can assign the correct (w.r.t. the part-wholetaxonomy) domain & range restrictions to the roles thus obtaining ontologically correct deductions14.

For illustration, let us assume the user chooses option (i) instead of changing the domain and rangerestrictions. The result in the RBox is that the roles are inverted such that now part-of v involved-in, whichcontradicts the part-whole taxonomy, but is logically correct. Test 2 and 3 then pass. Subsequent check-ing of the TBox (A1) will still yield an inconsistent Chassis, because the RBox is ontologically flawed.More precisely, Chassis is part-of Car and both are still subconcepts of ED, whereas part-of is typed with PDthat is disjoint from ED; hence, Chassis as part of Car can never be instantiated.

Going along with option (iii) can, in this example, still yield an inconsistent theory. Choosing (iii), thefollowing steps occur. A new axiom for involved-in is proposed for the domain restriction: with D′

R ≡DR u DS we have in the example D′

R ≡ PT u PD, which, with PD v PT, results in D′R ≡ PD, and

upon accepting this proposal we get involved-in v PD × PT. The same sequence is repeated for the rangerestrictions, such that we have involved-in v PD × PD. Then Test 2 and 3 can be executed. Recollecting,we have: part-ofv PD× PD and involved-inv part-of. Thus, the compatibility reasoning service will suggestoptions (a) and (b) to the user. By choosing option (b) the user has the possibility to change the domainand range restrictions to those in RBox (B). If, on the other hand, the users accepts the option (a) bothconcepts Chassis and Car will be unsatisfiable.

Looking briefly at the combination (A2+C), then upon choosing option (i), both Chassis and Car arere-classified from PT to PD to comply with the domain & range restriction of part-of. Choosing option(iii) together with option (a) will still result into a re-classification of both Chassis and Car as subconceptsof PD. Although the conceptual model (or domain ontology) with the re-classification is satisfiable, itis ontologically flawed both regarding the role hierarchy and the concept hierarchy, and again, selectingoption (ii) to correct the domain & range restrictions to that of RBox (B) is the appropriate choice. ♦

Thus, with the currently available reasoners, one may get error messages about inconsistent concepts orundesired equivalences and/or subsumptions between concepts, whereas the error is in the role hierarchy.The RBox compatibility service can be used equally after updating an ontology, and additional user-

14Although the precise definition of ‘ontologically correct’ is a topic of active research efforts, it is thus far generally used tocontrast it with obvious modeling errors (is_a vs part_of, mixing criteria for subsumption) and corrections made thanks to, e.g.,the OntoClean methodology (Guarino and Welty, 2004) or disambiguation of relations through usage of the Relation Ontology(Smith et al., 2005).

16 C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations

friendly messages might be devised for the user interface of the CASE (or ontology development) tools toexplain the five choices of the three tests.

Relations/properties/roles are essential components in both conceptual models and ontologies, whichreceives a more prominent place when role hierarchies and domain & range restrictions are properlydeclared, as with the part-whole relations taxonomy. To reason over the conceptual models and domainontologies requires inclusion of reasoning over both concepts and roles to check if they are consistent. Weshowed how ontologically unwanted logical implications concerning the concept hierarchy can be avoidedwith the additional compatibility check on the role hierarchy. These aspects, in turn, ease conceptualmodelling and domain ontology development and results in a representation that is closer to the real worldsemantics it intends to represent.

6. Conclusions and Further Research

We have introduced a formal taxonomy of part-whole relations that are commonly used in conceptualmodelling. The main rationale for distinguishing types of part-whole relations are (in)transitivity of the re-lation and the categories of the entity types they relate. This enables conceptual modellers (and developersof domain ontologies) to create representations that are closer to the real-world semantics, hence, improvequality of the software. In addition to addressing the problem of identifying different types of part-wholerelations, automated reasoning for using the taxonomy was investigated. This resulted in a new require-ment and corresponding reasoning service, called RBox compatibility, for automated reasoners. The RBoxcompatibility service checks the logical and ontological consistency of role domain & range restrictionswith respect to both the role hierarchy and the class hierarchy. The notion of RBox compatibility givesalso rise to formally introduce a notion of ontological correctness that can be distinct from the classicalnotion of logical consistency.

There are, however, multiple other facets when representing part-whole relations in conceptual mod-els, which we did not address here. These aspects include, among others, properties such as degree ofshareability (Motschnig-Pitrik and Kaasbøll (1999)), separability (Guizzardi (2005)), full support of morecomprehensive mereological theories, distributivity of properties (Artale et al. (1996)), and reasoning sce-narios with part-whole relations (e.g. Lambrix and Padgham, 2000). Methodologically, these facets needattention in the modelling process after identifying the most appropriate type of part-whole relation andafter the basic consistency checks over the relations and the classes have been run. We are currently in-vestigating trade-offs between comprehensiveness of representing more aspects of part-whole relationsversus expressivity of several description logic languages.

References

Artale, A., Franconi, E., Guarino, N., and Pazzi, L. (1996). Part-Whole Relations in Object-Centered Systems: an Overview.Data & Knowledge Engineering, 20(3), 347-383.

Artale, A., Franconi, E., Mandreoli, F. (2003). Description logics for modelling dynamic information. In Chomicki, J., van derMeyden, R., Saake, G. (Eds), Logics for Emerging Applications of Databases. Lecture Notes in Computer Science, Springer-Verlag, Berlin.

Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. F. editors. (2002). Description Logic Handbook:Theory, Implementation and Applications. Cambridge University Press, Cambridge.

Barbier, F., Henderson-Sellers, B., Le Parc-Lacayrelle, A., Bruel, J.-M. (2003). Formalization of the whole-part relationship inthe Unified Modelling Language. IEEE Transactions on Software Engineering, 29(5), 459-470.

Berardi, D., Calvanese, D., De Giacomo, G. (2005). Reasoning on UML class diagrams. Artificial Intelligence, 168(1-2), 70-118.Bittner, T., Donnelly, M. (2005). Computational ontologies of parthood, componenthood, and containment, In: Proceedings of

the Nineteenth International Joint Conference on Artificial Intelligence 2005 (IJCAI05). Kaelbling, L. (ed.). pp382-387.Borgo, S., Masolo, C. (2007). Full mereogeometries. Journal of Philosophical Logic. (to appear).Calvanese, D., Lenzerini, M., Nardi, D. (1998). Description logics for conceptual data modeling. In J. Chomicki and G. Saake,

editors, Logics for Databases and Information Systems. Kluwer, Amsterdam.Calvanese, D., De Giacomo, G., Lenzerini, M. (1999). Reasoning in expressive description logics with fixpoints based on au-

tomata on infinite trees. In: Proc. of the 16th Int. Joint Conf. on Artificial Intelligence (IJCAI’99), pp84-89.

C.M. Keet and A. Artale / Representing and Reasoning over a Taxonomy of Part-Whole Relations 17

Calvanese, D., De Giacomo, G. (2003). Expressive description logics. In: The Description Logic Handbook: Theory, Imple-mentation and Applications, Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P. (Eds). CambridgeUniversity Press, Cambridge. pp178-218.

Calvanese, D., De Giacomo, G., Lembo, D., Lenzerini, M., Rosati, R. (2005). DL-Lite: Tractable description logics for ontolo-gies. In: Proc. of the 20th Nat. Conf. on Artificial Intelligence (AAAI 2005), pp602-607.

Franconi, E., Ng, G. (2000). The iCom tool for intelligent conceptual modelling. 7th Intl Workshop on Knowledge Representationmeets Databases. Berlin, Germany.

Gerstl, P., Pribbenow, S. (1995). Midwinters, end games, and body parts: a classification of part-whole relations. InternationalJournal of Human-Computer Studies, 43, 865-889.

Guarino, N., Welty, C.A. (2004). An overview of OntoClean. In: Handbook on ontologies. Stab, S., Studer, R. (eds.). SpringerVerlag, 2004. pp151-159.

Guizzardi, G. (2005). Ontological foundations for structural conceptual models. PhD Thesis, Telematica Institute, Twente Uni-versity, Enschede, the Netherlands. ISBN 90-75176-81-3.

Herre, H., Heller, B. (2006). Semantic foundations of medical information systems based on top-level ontologies. Knowledge-Based Systems, 19, 107-115.

Horrocks, I., Kutz, O., Sattler, U. (2006). The Even More Irresistible SROIQ. In: Proceedings of the 10th International Con-ference of Knowledge Representation and Reasoning (KR-2006), Lake District, UK.

Johansson, I. (2004). On the transitivity of the parthood relation. In: Relations and predicates, Hochberg, H., Mulligan, K. (eds.).Ontos Verlag, Frankfurt. pp161-181.

Keet, C.M. (2006a). Part-whole relations in Object-Role Models. 2nd International Workshop on Object-Role Modelling (ORM2006), Montpellier, France, Nov 2-3, 2006. In: OTM Workshops 2006. Meersman, R., Tari, Z., Herrero., P. et al. (Eds.),Lecture Notes in Computer Science 4278. Berlin: Springer-Verlag, Berlin. pp1116-1127.

Keet, C.M. (2006b). Introduction to part-whole relations: mereology, conceptual modelling and mathematical aspects. KRDBResearch Centre Technical Report KRDB06-3 (Tutorial reader), Faculty of Computer Science, Free University of Bozen-Bolzano, Italy. 2 October 2006.

Keet, C.M. (2007). Mapping the Object-Role Modeling language ORM2 into Description Logic language DLRifd . KRDB Re-search Centre Technical Report KRDB07-2, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy. 15February 2007. arXiv:cs.LO/0702089v1.

Lambrix, P., Padgham, L. (2000). Conceptual modeling in a document management environment using part-of reasoning indescription logics. Data & Knowledge Engineering, 32(1), 51-86.

Masolo, C., Borgo, S., Gangemi, A., Guarino, N. and Oltramari, A. (2003). Ontology Library. WonderWeb Deliverable D18 (ver.1.0, 31-12-2003). http://wonderweb.semanticweb.org.

Masolo, C., Vieu, L., Bottazzi, E., Catenacci, C., Ferrario, R., Gangemi, A., Guarino, N. (2004). Social Roles and their Descrip-tions. 9th International Conference on Principles of Knowledge Representation and Reasoning (KR2004). Banff, Canada.

Motschnig-Pitrik, R., Kaasbøll, J. (1999). Part-Whole Relationship Categories and Their Application in Object-Oriented Analy-sis. IEEE Transactions on Knowledge and Data Engineering, 11(5), 779-797.

Object Management Group. (2005). Unified Modeling Language: Superstructure. v2.0. formal/05-07-04.http://www.omg.org/cgi-bin/doc?formal/05-07-04.

Odell, J.J. (1998). Six different kinds of composition. Advanced Object-Oriented Analysis & Design using UML. CambridgeUniversity Press, Cambridge.

Opdahl, A., Henderson-Sellers, B., Barbier F. (2001). Ontological Analysis of whole-part relationships in OO-models. Informa-tion and Software Technology, 43, 387-399.

Rector, A., Rogers, J., Bittner, T. (2006). Granularity, scale and collectivity: When size does and does not matter. Journal ofBiomedical Informatics, 39(3), 333-349.

Sattler, U. (1995). A concept language for an engineering application with part-whole relations. In: Proceedings of the interna-tional workshop on description logics, Borgida, A., Lenzerini, M., Nardi, D., Nebel, B. (Eds.), pp119-123.

Schneider, L. (2003). Designing Foundational Ontologies. The Object-Centered High-level Reference Ontology OCHRE asa Case Study. In: Song, I.-Y.; Liddle, S.W.; Ling, T.W.; Scheuermann, P. (eds.), Conceptual Modeling - ER 2003, 22ndInternational Conference on Conceptual Modeling. LNCS 2813. Heidelberg: Springer, 91-104.

Schulz, S., Hahn, U., Romacker, M. (2000). Modeling Anatomical Spatial Relations with Description Logics. Proceedings of theAMIA Symposium 2000. Overhage, J.M. (ed.), pp779-83.

Schulz, S., Kumar, A., Bittner, T. (2006). Biomedical ontologies: What part-of is and isn’t. Journal of Biomedical Informatics,39(3), 350-361.

Shanks, G., Tansley, E., Weber, R. (2004). Representing composites in conceptual modeling. Communications of the ACM, 47(7),77-80.

Simons, P. (1987). Parts: a study in Ontology. Clarendon Press, Oxford.Smith, B., Ceusters, W., Klagges, B., Köhler, J., Kumar, A., Lomax, J., Mungall, C.J., Neuhaus, F., Rector, A., Rosse, C. (2005).

Relations in Biomedical Ontologies. Genome Biology, 6, R46.Tan, H.B.K, Hao, L., Yang, Y. (2003). On formalizations of the whole-part relationship in the Unified Modeling Language. IEEE

Transactions on Software Engineering, 29(11), 1054-1055.Varzi, A.C. (2004). Mereology. The Stanford Encyclopedia of Philosophy. (Fall 2004 Edition), Zalta, E.N. (ed.).

http://plato.stanford.edu/archives/fall2004/entries/mereology/.Varzi, A.C. (2006a). Spatial reasoning and ontology: parts, wholes, and locations. In: The Logic of Space, Aiello,

M., Pratt-Hartmann I., van Benthem J. (eds.). Dordrecht: Kluwer Academic Publishers. (to appear) http://www.columbia.edu/∼av72/papers/Space_2006.pdf, date accessed: 18-9-2006.

Varzi, A.C. (2006b). A Note on the Transitivity of Parthood. Applied Ontology (to appear).Vieu, L., Aurnague, M. (2005). Part-of Relations, Functionality and Dependence. In: M. Aurnague, M. Hickmann, L. Vieu (eds.),

Categorization of Spatial Entities in Language and Cognition. John Benjamins, Amsterdam.Winston, M.E., Chaffin, R., Herrmann, D. (1987). A taxonomy of part-whole relations. Cognitive Science, 11(4), 417-444.